D. Jo{t - L. [kerget: Napoved izkoristka - Predict the Efficiency
V prispevku je predstavljena numerièna analiza toka v francisovi turbini. Osredotoèili smo se na napoved energijskih izgub v toku in napoved izkoristka turbine. Rezultate, dobljene z loèeno analizo toka v vsakem delu turbine, smo primerjali z rezultati loèenega izraèuna toka v spirali in skupnega izraèuna toka skozi preostalo turbino. Nato smo izraèunali tok v francisovi turbini v veèjem tevilu obratovalnih toèk, narisali koljèni diagram izkoristka in ga primerjali z izmerjenim. Enak izraèun je bil narejen e za primer, ko nimamo izmerjenih vstopnih podatkov. V prispevku je predstavljen tudi vpliv gostote mree in izbire turbulentnega modela na rezultate.
© 2001 Strojniki vestnik. Vse pravice pridrane.
(Kljuène besede: turbine francisove, izkoristek turbin, analize toka, modeli turbulentni )
This paper presents a numerical analysis of flow in a Francis turbine. We concentrate on flow-energy losses and efficiency prediction. The results, obtained by a separate analysis of each turbine component, are compared with the results of a separate analysis of flow in a spiral casing and a simultaneous calculation of the flow through the other turbine parts. After this we analysed flow in a Francis turbine at several operating points, an efficiency hill-chart diagram was drawn and compared with the measured one. The same calcula-tion was made for a case where no measured inlet condicalcula-tions were avaible. The effect of the grid density and turbulence models on the results is also presented.
© 2001 Journal of Mechanical Engineering. All rights reserved.
(Keywords: Francis turbine, turbine efficiency, flow analysis, turbulence models)
Using Numerical Flow Analysis to Predict the Efficiency of a Francis Turbine
Dragica Jo{t - Leopold [kerget
Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01)
Napoved izkoristka francisove turbine z
numeri~nim izra~unom toka
0 UVOD
Od rezultatov numeriène analize toka v turbini prièakujemo natanèno informacijo o toku, primerno toèen izraèun tokovnih izgub in izkoristka in napoved kavitacije. Pri oblikovanju novih gonilnikov in drugih delov turbinskih strojev je numerièna analiza toka nepogreljivo orodje. Mnogo lae je oblikovati veliko tevilo lopatic gonilnika na raèunalniku in na podlagi numeriène analize izbrati najboljega, kakor pa izdelati tevilne modele in z meritvami izkoristka izbrati najboljega. Pomembno pa je, da na podlagi numeriènih rezultatov res izberemo najbolji gonilnik. Bolj kot absolutna vrednost numerièno dobljenega izkoristka nas zanimata lega optimalne toèke obratovanja in oblika diagrama izkoristka.
V preteklosti smo raèunali vsak del turbine posebej. Rezultate analize toka skozi en del turbine smo uporabili za vstopne pogoje pri analizi naslednjega dela. Tak izraèun ne upoteva vpliva
0 INTRODUCTION
Numerical results are expected to give de-tailed information about the flow in a turbine; to pre-dict flow-energy lossses and efficiency with reason-able accuaracy; and to foresee the cavitation. In the design process of runners and other turbine compo-nents CFD is a useful tool: it is much easier to design a number of runner blades on a computer and nu-merically choose the best one than to do several models and model tests. But it is essential to choose the best runner. More than the absolute value of effi-ciency, it is important to accurately obtain the posi-tion of the best-efficency point and the shape of the efficiency diagram.
previ-ene komponente turbine na poprejnjo. Tako izgubimo vpliv gonilnika na tok v dvojni kaskadi in vpliv sesalne cevi na tok v gonilniku. Kljub temu pa so bili rezultati loèene analize toka pogosto uspeno uporabljeni pri izboljanju hidravliènih oblik vseh delov turbine ([1] in [2]).
V zadnjih letih je bil v numeriènem obravnavanju toka tekoèin doseen izreden napredek. Eden najpomembnejih dosekov je skupni izraèun toka v rotirajoèih in nerotirajoèih delih stroja. Tako je zdaj mogoèe skupaj raèunati tok od vstopa v spiralo do izstopa iz sesalne cevi z vsemi predvodilnimi, vodilnimi in gonilnimi lopaticami. Tako upotevamo medsebojni vpliv statorja, gonilnika in sesalne cevi in se izognemo nenatanènim robnim pogojem med komponentami. Slaba stran takega izraèuna je veliko tevilo vozlov, poèasna konvergenca in dolgi raèunski èasi. Kompromisna reitev je loèen izraèun toka v spirali in skupen izraèun toka od vstopa v predvodilnik do izstopa iz sesalne cevi, obmoèja raèunanja pri kaskadah predvodilnika, vodilnika in gonilnika pa so skrèena na en perodièni del ([3] in [4]). Pogosto pa je tudi tak izraèun prezamuden in se moramo odloèiti za loèeno analizo toka. Zanesljivost numeriènih rezultatov je odvisna tudi od gostote mree. V primeru premajhnih raèunalnikih zmogljivosti je vpraanje, ali je bolje raèunati celotno turbino na redki mrei ali pa vsak del turbine posebej na zgoèeni mrei. Rezultati so odvisni tudi od izbire turbulentnega modela.
V tem prispevku skuamo prikazati razlike med rezultati loèenega in skupnega izraèuna, vpliv gostote mree in izbire turbulentnega modela na rezultate in zanesljivost numeriènega izraèuna izkoristka turbine.
1 LOÈENA, DELNO SKLOPLJENA IN SKLOPLJENA ANALIZA TOKA
Numerièna analiza toka je bila narejena za model francisove turbine s specifièno vrtilno frekvenco nS=300 (ns=3.65 nQ1/2 H-3/4), ki je bila
izmerjena na Turbointitutu. Turbina sestoji iz spirale z 12 predvodilnimi lopaticami, 24 vodilnih lopatic, iz 13-lopatiènega gonilnika in kolenaste sesalne cevi z navpiènim rebrom v izstopnem delu.
Numerièna analiza toka je bila narejena s programskim paketom CFX-TASCflow s standardnim modelom k - e.
Obratovalna toèka turbine je doloèena s padcem, pretokom in vrtljaji. Namesto pretoka in padca raje uporabljamo pretoèno tevilo j (j=Q/(pwr3)) in tlaèno
tevilo y (y=2gH/(wr)2). Tu sta j in y brezdimenzijski
tevili in sta neodvisni od velikosti stroja. Numerièna analiza toka je bila narejena za pet obratovalnih toèk pri nominalnem y. Pretok pri doloèenem odprtju vodilnika in vrtljajih je bil dobljen iz meritev.
ous one. So the influence of a runner on the flow through the distributor and the influence of a draft tube on the flow in the runner were lost. In spite of this, results of separate analyses were used success-fully to improve the hydraulic shapes of all turbine parts ([1] and [2]).
Recently, there has been a rapid develop-ment in CFD and one of the most important achieve-ments is simultaneous calculation of the flow in rotat-ing and non-rotatrotat-ing parts. It is now possible to calcu-late the flow from the spiral casing inlet to the draft-tube outlet with all the stay and guide vanes and the runner blades, simultaneously. In this way we take into account the interaction of the stator, the rotor and the draft tube and avoid inaccurate boundary condi-tions between the turbine components. The disad-vantages of such a calculation are the large number of nodes, the slow convergence and the long CPU time. The compromise solution is a separate analysis of the spiral casing and a simultaneous calculation of the flow from the stay-vanes inlet to the draft-tube outlet, while the domain for the stay and guide vanes and the runner-blades cascades is reduced to one periodic part ([3] and [4]). Often, even this kind of calculation is too time consumming and a separate analysis has to be performed. The reliability of the numerical results also depends on the grid density. In the case of insufficient computer capacity there is a question as to whether it is better to calculate the whole turbine on a coarse grid or to calculate each part individualy on a fine grid. The results also depend on the turbulence model.
In this paper the difference between the re-sults of separate and coupled analysis, the effect of grid density and the turbulence model on the results and the reliability of numerically predicted efficiency are presented.
1 SEPARATED, PARTLY COUPLED AND COUPLED-FLOW ANALYSIS
A numerical analysis was made for a model of a Francis turbine with specific speed nS=300,
(ns=3.65 n Q1/2 H-3/4), which was tested on the test rig
at the Turboinstitute. The turbine consists of a spiral casing with 12 stay vanes, 24 guide vanes, a 13-blades runner and an elbow draft tube with a vertical pier.
The numerical analysis was made with the CFX-TASCflow computer code using the standard k - e model.
The turbine operating point is determined by head, discharge and speed. Often, instead of discharge and head, a discharge coefficient j (j=Q/(pwr3)) and a
pressure coefficient y (y=2gH/(wr)2) are used. Here j
Najprej je bila narejena analiza toka v spirali s predvodilnimi lopaticami. Obmoèje raèunanja je razirjeno do izstopa iz gonilnika, vendar brez vodilnih in gonilnih lopatic. V mrei je 280 000 vozlov (sl. 1). Iz rezultatov izraèuna toka v spirali smo dobili vstopne pogoje za nadaljnje izraèune. Izraèunali smo tudi izgube v spirali.
Numerièna analiza toka v dvojni kaskadi, gonilniku in sesalni cevi je bila narejena na tri naèine. Prvi naèin je bil loèen izraèun toka v kaskadi, gonilniku in sesalni cevi. Obmoèje raèunanja za dvojno kaskado je del med dvema predvodilnima in tremi vodilnimi lopaticami in med vencem in pestom, toda brez lopatic gonilnika. V mrei je 114 000 vozlov. Obmoèje raèunanja za gonilnik je med dvema lopaticama, v mrei je 72 000 vozlov. Mrea za sesalno cev vsebuje 170 000 vozlov. Vstopni pogoji so dobljeni iz analize prejnje komponente. Obmoèja raèunanja se prekrivajo, ker smo eleli zmanjati vpliv nenatanènih izstopnih robnih pogojev. Drugi naèin je delno sklopljena analiza toka. Tok skozi predvodilne in vodilne lopatice ter gonilnik raèunamo skupaj (166 000 vozlov), tok v sesalni cevi pa posebej (170 000 vozlov). Tretji naèin je skupen izraèun toka od vstopa v predvodilnik do izstopa iz sesalne cevi. Mrea vsebuje 332 000 vozlov (sl. 2). Mree za sklopljeno analizo so bile dobljene z zdruevanjem mre loèenega izraèuna, izpuèeni so bili le deli, ki se prekrivajo. Zato je struktura in gostota mree enaka za loèen in sklopljen izraèun.
Iz rezultatov numeriènega izraèuna lahko izraèunamo izgube v toku, navor na os turbine in izkoristek. V nerotirajoèih delih turbine razlika med totalnim tlakom na vstopu in izstopu pomeni izgube:
pri èemer je
vt je transportna komponenta hitrosti, S1 in S2 pa
vstopni in izstopni prerez. Èe DE delimo s tenostnim pospekom g, dobimo izgube, izraene kot del padca, ki ni bil izkorièen. V gonilniku veèino razlike v totalnem tlaku pomeni delo gonilnika, majhen del pa izgube v toku. Izkoristek gonilnika izraèunamo po obrazcu:
pri èemer je M navor na os turbine, w pa kotna hitrost. V primeru loèene numeriène analize dobimo DE kot vsoto prispevkov posameznih komponent.
First, a numerical analysis of the spiral cas-ing with stay vanes was performed. The computa-tional domain was extended to the runner outlet, but the guide vanes and runner blades were not mod-eled. The grid consisted of 280 000 nodes (Fig. 1). The results were used as the inlet conditions for sub-sequent calculations. At the same time the flow-en-ergy losses in the spiral casing were calculated.
A numerical analysis of the flow in the tandem cascade, the runner and the draft tube was made in three stages. The first stage was a separate analysis of the flow through the tandem cascade, the runner and the draft tube. The computational domain for the tandem cascade is the region between two stay vanes and three guide vanes and between the hub and crown, but without runner blades. The domain consists of 114 000 nodes. The computational domain for the runner analysis is the region between two blades, it consists of 72 000 nodes. In the draft tube there are 170 000 nodes. The inlet conditions were obtained from numerical results of the upstream component. In or-der to minimize the influence of the inaccurate outlet bound-ary conditions the computational domains overlapped. The second stage was a partly coupled analysis. Flow through the stay vanes, the guide vanes and the runner were calcu-lated simultaneously (166 000 nodes), while the draft tube was analysed separately (170 000 nodes). Finally, the flow from the stay-vanes inlet to the draft-tube outlet was analysed simultaneously. The grid consisted of 332 000 nodes (Fig. 2). The grids for the coupled analysis were obtained by attaching the grids of separate analyses and omitting the parts which overlapped, so the grid structure and grid density were the same for the separate and coupled analyses.
From the numerical results the flow-energy losses, the torque on the shaft and the efficiency can be calculated. Flow-energy losses in the non-rotating turbine parts are calculated as the difference between the total pressure at the domain inlet and outlet
(1),
where
(2),
vt is transport velocity component, S1 and S2 are the inlet
and outlet cross-sections. If DE is divided by the accel-eration due to gravity g, the flow-energy losses can be expressed as a head, which was not utilized. In the runner most of the difference in total pressure is converted to runner work, while a small part represents flow-energy losses. The turbine efficiency can be calculated by:
(3),
where M is the torque on the shaft, and w is the angular velocity. In the case of a separate numerical analysis DE is obtained as the sum of the contributions of all the turbine parts.
1 2
1
. tot t tot t
S S
E p v dS p v dS
Q r
æ ö
D = çç - ÷÷
è
ò
ò
ø2
1 . 2 tot
p = rv +p
Podroben prikaz rezultatov loèene, delno sklopljene in sklopljene analize je prikazan v [5]. Pokazalo se je, da loèena analiza toka napove prevelike izgube v vseh delih turbine, izraèunani navor na os turbine pa je skoraj enak pri loèeni, delno sklopljeni in sklopljeni analizi toka. Zato loèena analiza toka napove bistveno manji izkoristek, kakor je bil izmerjen. Tudi lega optimalne toèke obratovanja je pomaknjena k veèjemu pretoku. Rezultati delno sklopljenega izraèuna so nekoliko blie izmerjenim. Oblika diagrama izkoristka, dobljenega s sklopljenim izraèunom, se dobro ujema z izmerjenim, vrednosti izkoristka pa so za okoli 3% manje od izmerjenih (sl. 3).
2 KOLJÈNI DIAGRAM IZKORISTKA
Na podlagi rezultatov za nominalno tlaèno tevilo smo ugotovili, da je le sklopljena analiza primerna za izraèun izkoristka turbine. Zato smo s sklopljeno analizo izraèunali izkoristek v naslednjih
A detailed comparison of the results of the separate, the partly coupled and the coupled analysis is presented in [5]. We found that the separate flow analysis overestimates the flow-energy losses in all the turbine parts, while the calculated torque on the shaft is nearly the same for the separated, the partly coupled and the coupled calculations. In other words, the sepa-rate analysis predicts a much lower efficiency than the measured value. Also, the position of the best-efficiency point is shifted to a higher discharge. The results of the partly coupled calculation are closer to the measured values. The shape of the efficiency curve obtained with the coupled analysis is in good agreement with the mea-sured value, but the calculated efficiency is about 3% lower than the measured values (Fig. 3).
2 HILL-CHART DIAGRAM
On the basis of the results for the nominal pressure coefficient it was concluded that only the cou-pled analysis is suitable for the prediction of turbine efficiency. Therefore, only the coupled analysis was Sl. 1. Obmoèje raèunanja in mrea za izraèun toka v spirali
Fig. 1. Computational domain and grid for flow analysis in the spiral casing
Sl. 2. Obmoèje raèunanja pri skupnem izraèunu toka skozi predvodilnik, vodilnik, gonilnik in sesalno cev Fig. 2. Computational domain for simultaneous calculation of the flow through the stay and guide vanes,
desetih toèkah obratovanja, pet za nije in pet za vije tlaèno tevilo. Izraèunane in izmerjene krivulje izkoristka se po obliki dobro ujemajo, izraèunani izkoristek je za okoli 3% manji, le pri velikem pretoènem tevilu in majhnem tlaènem tevilu (j=0,2336, y=0,8575), smo dobili odstopanje okoli 5%.
Na temelju izraèunanega izkoristka v 15 obratovalnih toèkah nariemo koljèni diagram izkoristka. Za primerjavo nariemo e diagram na podlagi izmerjenega izkoristka v 15 obratovalnih toèkah (sl. 4). Numeriène vrednosti izkoristka so deljene z najveèjim izraèunanim izkoristkom, izmerjene vrednosti pa z najveèjim izmerjenim izkoristkom. Obravnavane toèke obratovanja so na preseèièih krivulj, ki pomenijo nespremenljivo odprtje vodilnika in vodoravnih èrt, ki pomenijo stalen y. Lega optimalne toèke obratovanja se dobro ujema z meritvami, prav tako tudi oblika krivulj s stalnim izkoristkom. Izraèunani diagram je v okolici optimalne toèke obratovanja nekoliko bolj poloen, dlje od optimalne toèke obratovanja pa bolj strmo pade kakor izmerjeni. Razlika je najveèja v desnem spodnjem delu diagrama, zaradi veèjega odstopanja med izmerjenim in izraèunanim izkoristkom pri
j=0,2336, y=0,8575.
made for the additional ten operating points, five for the lower and five for the higher pressure coefficient. The calculated and measured efficiency curves have the same shape, however, the calculated efficiency is about 3% lower. The exception is the operating point, with a large discharge coefficient and a small pressure coefficient (j=0.2336, y=0.8575), where the discrepancy is 5%.
On the basis of the calculated efficiency for the 15 operating points a hill-chart diagram was drawn. For comparison, a diagram based on the measured effi-ciency at 15 operating points was also drawn (Fig. 4). The numerically obtained values were divided by the highest calculated efficiency, while the experimental values were divided by the highest measured efficiency. The treated operating points were at the cross-sections of the curves of constant guide-vane opening (A0) and
the lines of constant y. The position of the best-effi-ciency point was quite accurately predicted. The shape of the efficiency contours was also in quite good agree-ment. Near the best-efficiency point the calculated dia-gram is flatter than the measured one, but further from the best-efficiency point the efficiency decreases quickly. The descrepancy is largest at the bottom right-hand part of the diagram, because of the larger disa-greement between the measured and calculated effi-ciency at point j=0.2336, y=0.8575.
Sl. 4. koljèna diagrama izkoristka na temelju izraèunanih in izmerjenih vrednosti v 15 toèkah
Fig. 4. Hill-chart efficiency diagram based on the calculated and measured efficiency at 15 points
0,8 0,85 0,9 0,95 1
0,17 0,19 0,21 0,23 0,25 0,27
j
h
/hop
t
1
2
3
0
Sl. 3. Diagram izkoristka za nominalno tlaèno tevilo y=1,011
1 - loèen izraèun, 2 - delno sklopljen izraèun, 3 - sklopljen izraèun, 0 - meritve Fig. 3. Efficiency diagram for the nominal pressure coefficient y=1.011
3 THE EFFECT OF GRID DENSITY AND TURBU-LENCE MODELS ON THE RESULTS
The efficiency diagram for a nominal pres-sure coefficent was obtained from the results of a numerical analysis performed by the standard k-e
model on coarse grids. In order to study the effect of the grid density and the turbulence model on the results, a separate numerical analysis at the best-efficiency point (BEP) was performed using different turbulence models for the coarse and refined grids.
The pressure and velocity distribution obtained for grids of different density are qualitatively similar. For the case of coarse grids the flow-energy losses are too high, mostly due to overprediction of the friction losses. With grid refinement the flow-energy losses decrease, especially in the tandem cascade and the runner, while in the draft tube the effect is small. The same effect of grid refinement was obtained with all the turbulence models.
3.1 Turbulence models
The calculations were made with two-equational models k-e and k-w. Besides the standard k-e model the RNG model was also used. This model is obtained from Renormalized Group Theory applied to Navier-Stokes equations. The transport equations for k and e are the same as for the case of the standard
k-e model, but the closure coefficients are different [6]. The k-w turbulence model was developed to predict the onset of separation on a smooth surface more accurately. In CFX-TASCflow three k-w models are available: the standard Wilcox model [7], the BSL (Baseline) model and the SST model (Shear Stress Trans-port model). The standard k-w model is very sensitive to the inlet conditions for w. The BSL model combines the advantages of the k-e and k-w models. The Wilcox model is multiplied by a blending function F1. The k-e
model is at first transformed to the k-w formulation and then multiplied by (1-F1). F1 is defined in such a way that
outside the boundary layer the standard k-e model is used, while inside the boundary layer the k-w model is used [8]. The SST model accounts for the transport of the turbulent shear stress and therefore predicts the separation most accurately [9].
One of the problems with the two- equational models is the behavior near stagnation points. It is frequently observed that very high turbulence levels are predicted upstream of a stagnation point and then transformed around the body. This problem was solved by Kato and Launder, who changed the production term in the equation for the turbulent kinetic energy [10].
When the k-e turbulence model is used, the flow in the near-wall region can be modeled by standard log-law wall functions. To avoid inconsistencies for the case of fine grids, fixed y+
wall functions were developed [11]. For the k-w model 3 VPLIV GOSTOTE MREE IN TURBULENTNIH
MODELOV NA REZULTATE
Diagram izkoristka za nominalno tlaèno tevilo je bil dobljen iz rezultatov numeriène analize, izvedene s standardnim modelom k-e na precej redki mrei. Da bi preuèili vpliv gostote mree in turbulentnega modela na rezultate, je bila loèena analiza toka v optimalni toèki obratovanja narejena z razliènimi turbulentnimi modeli na redki in zgoèeni mrei.
Porazdelitve tlaka in hitrosti, dobljene na mreah razliène gostote, so kakovostno zelo podobne. Pri redkih mreah so energijske izgube v toku veèje predvsem na raèun precenjenih izgub zaradi trenja na stenah. Z zgostitvijo mre se izgube zmanjajo, zlasti v dvojni kaskadi in v gonilniku, medtem ko je vpliv zgostitve mree v sesalni cevi manji. Vpliv zgostitve mree je enak za vse uporabljene turbulentne modele.
3.1 Turbulentni modeli
Pri izbiri turbulentnih modelov smo se omejili na dvoenaèbna modela k-e in k-w. Poleg standardnega modela k-e smo raèunali tudi z modelom RNG. Ta model je dobljen s teorijo renormalizacijskih grup (RNG), uporabljeni na Navier-Stokesovih enaèbah. Transportni enaèbi za k in e sta enaki kakor pri standardnem modelu k-e, razlikujejo se le koeficienti, s katerimi sklenemo sistem [6].
Turbulentni model k-w je bil razvit z namenom, da bi bolj natanèno napovedali odlepljanje toka na gladkih stenah. V CFX-TASCflow so vkljuèeni trije modeli k-w: standardni Wilcoxov model k-w [7], model BSL (Baseline model) in model SST (Shear Stress Transport model). Standardni model k-w je zelo obèutljiv za vstopne pogoje za w. Model BSL skua ohraniti prednosti modelov k-e in k-w. Pri tem modelu je Wilcoxov model pomnoen s funkcijo F1, model
k-e pa je najprej transformiran v obliko k-w, nato pa pomnoen s (1-F1). F1 je definirana tako, da imamo
zunaj mejne plasti standardni model k-e, ob steni pa preidemo na model k-w [8]. Model SST upoteva prenos turbulentnih strinih napetosti in najbolje popie odlepljanje toka na stenah [9].
Eden od problemov dvoenaèbnih modelov je obnaanje v okolici zastojnih toèk. Pogosto opazimo pred zastojnimi toèkami zelo visok nivo turbulence, ki se nato porazdeli okoli telesa. Pro-blem sta reila Kato in Launder s spremembo produkcijskega èlena v enaèbi za turbulentno kinetièno energijo [10].
the problem of inconsistencies in the case of fine grids is solved by a formulation which automatically switches from wall functions to a low-Re near-wall formulations as the grid is refined [12].
In CFX-TASCflow a two-layer turbulence model is also available. The standard k-e model is used away from the wall, while the one-equation model is used near the wall. For high Reynolds numbers a very fine grid near the walls is required [12]. Due to insufficient computer capacity, we did not use this model.
3.2 Flow calculation with different turbulence models
Flow in the tandem cascade was calculated with the standard and the RNG k-e models. With the RNG model, smaller flow-energy losses were obtained than with the standard k-e model. The calculation was repeated with standard k-w and SST k-w models. With the stan-dard k-w model, due to larger wakes behind the vanes, slightly larger flow-energy losses were obtained than with the standard k-e model. The distribution of turbulent kinetic energy, obtained with the different models on the refined grid is presented in Fig. 5. With the standard k-e
(Fig. 5a) and the standard k-w (Fig. 5c) models, an in-crease in the turbulent kinetic energy near the stagnation problem nedoslednosti reen s formulacijo, ki pri
zgoèenih mreah avtomatièno preide iz stenskih funkcij za visoka Re tevila na model za nizka Re tevila [12].
V CFX-TASCflow je vkljuèen tudi dvoslojni turbulentni model. Pri tem modelu tok dovolj stran od sten modeliramo z modelom k-e, tok ob stenah pa z enoenaèbnim modelom. Za visoka Reynoldsova tevila mora biti ob stenah mrea zelo zgoèena [12]. Zaradi premajhnih raèunalnikih zmogljivosti tega modela nismo uporabili.
3.2 Izraèun toka z razliènimi turbulentnimi modeli
Tok v dvojni kaskadi smo raèunali s standardnim in z modelom RNG k-e. Z modelom RNG dobimo nekoliko manje energijske izgube v toku kakor s standardnim modelom. Izraèun je bil ponovljen s standardnima modeloma k-w in SST
k-w. S standardnim modelom k-w dobimo predvsem zaradi nekoliko veèje sence za lopaticani za malenkost veèje izgube kakor s standardnim modelom k-e. Na sliki 5 je prikazana porazdelitev turbulentne kinetiène energije, dobljene z razliènimi modeli na gosti mrei. S standardnima modeloma k-e (sl. 5a) in k-w (sl. 5c) modeloma dobimo veliko
Sl. 5. Porazdelitev turbulentne kinetiène energije v dvojni kaskadi Fig. 5. Distribution of the turbulent kinetic energy in the tandem cascade
points and around the vanes can be observed. When using the Kato-Launder production term the distribution of turbulent kinetic energy is much more uniform (Fig. 5b and 5d). This results in a reduction of the flow-energy losses in the case of the k-e model by 2.5% on the coarse grid and 4% on the refined grid, while in the case of the
k-w model the reduction was 4% on the coarse grid and 9.7% on the refined grid. The smallest flow-energy losses were obtained with the SST k-w model, where the distri-bution of turbulent kinetic energy was also the most uni-form. In the case of the k-e model some reduction in the flow-energy losses was also obtained by the use of wall functions with a fixed y+.
As a result of the runner analysis, the torque on the shaft and the runner efficiency were compared. The difference in torque on the shaft is small, less than 0.35%. The runner efficiency is the smallest in the case of the standard k-e model and the largest in the case of the k-w SST model. With grid refinement, the runner efficiency increases by approximately 1%.
The flow in the draft tube was calculated with both two-equational models. The flow-energy losses and the coefficient of pressure recovery (Cp) were
com-pared. Cp represents the ratio of the difference in
pres-sure at the draft-tube outlet and inlet and the kinetic energy at draft tube inlet. Cp is defined by the equation:
(5),
where S1 and S2 are the inlet and outlet draft-tube
cross-sections, respectively, p is the pressure, vt the transport
velocity component, v the absolute velocity. The flow-energy losses obtained with the k-e model are smaller zveèanje turbulentne kinetiène energije v okolici
zastojnih toèk in okoli lopatic. Pri obeh modelih s Kato-Launderjevim produkcijskim èlenom dobimo veliko bolj enakomerno porazdelitev turbulentne kinetiène energije (sl. 5b in 5d). Zato dobimo tudi manje izgube, in sicer pri modelu k-e na redki mrei za 2,5%, na gosti mrei pa za 4%, pri modelu k-w pa na redki mrei za 4%, na gosti mrei pa za 9,7%. Najmanje izgube dobimo z modelom SST k-w, tam je tudi porazdelitev turbulentne kinetiène energije najbolj enakomerna (sl. 5d). Pri modelu k-e se izgube nekoliko zmanjajo e z uporabo stenskih funkcij s fiksnim y+.
Pri izraèunu toka v gonilniku smo primerjali navor na os turbine in izkoristek gonilnika. Razlike v navoru so majhne, pod 0,35%. Izkoristek gonilnika je najmanji pri standardnem modelu k-e in najveèji pri modelu SST k-w. Z zgostitvijo mree se pri vseh modelih izkoristek poveèa za priblino 1%.
Tok v sesalni cevi smo raèunali z obema dvoenaèbnima modeloma. Primerjali smo izgube in koeficient rekuperacije tlaka Cp. Koeficient
rekuperacije tlaka predstavlja razmerje med razliko tlaka na izstopu in vstopu sesalne cevi in kinetièno energijo na vstopu. Definiran je z enaèbo:
pri èemer je S1 vstopni, S2 pa izstopni prerez sesalne
cevi, p je tlak, vt je transportna komponenta, v pa
absolutna hitrost. Z modelom k-e dobimo manje izgube in viji Cp kakor z modelom k-w. S
Kato-2 1
2
2
t t
S S
p
t pv dS pv dS C
v v dS r
-=
ò
ò
ò
Preglednica 1. Energijske izgube v toku v kaskadi, dobljene z razliènimi turbulentnimi modeli na redki in gosti mrei
Table 1. Flow-energy losses in the tandem cascade obtained by several turbulence models on coarse and refined grids
A - standardne log. stenske funkcije / standard log.-law wall functions B - stenske funkcije s fiksnim y+ / fixed y+ wall functions
C - kombinacija stenskih funkcij za nizka in visoka Re tevila / combined low and high Re wall functions
Turbulentni model
Turbulence model Model za tok ob steni Near-wall model
DE/g
m 113 580 vozlov
113 580 nodes
DE/g
m 374 130 vozlov
374 130 nodes
k-e, standardni model /k-e, standard model A 0,4289 0,3728 k-e, standardni model /k-e, standard model B 0,4114 0,3616
k-e, Kato-Launder A 0,4184 0,3578
k-e, Kato-Launder B 0,4054 0,3450
k-e, RNG A 0,4135 0,3590
k-e, RNG, Kato-Launder A 0,4023 0,3547
k-e, RNG, Kato-Launder B 0,4023 0,3662
k-w, standardni model /k-w, standard model C 0,4338 0,3828
k-w, Kato-Launder C 0,4163 0,3455
Preglednica 2. Navor na os turbine in izkoristek gonilnika, dobljena z razliènimi turbulentnimi modeli na redki in zgoèeni mrei
Table 2. Torque on the shaft and runner efficiency obtained by several turbulence models on coarse and refined grids
A - standardne log. stenske funkcije / standard log.-law wall functions B - stenske funkcije s fiksnim y+ / fixed y+ wall functions
C - kombinacija stenskih funkcij za nizka in visoka Re tevila / combined low and high Re wall functions
72 000 vozlov,
72 000 nodes 243 000 vozlov, 243 000 nodes Turbulentni model
Turbulence model Model za tok ob steni Near-wall model M
Nm % h Nm M % h
k-e, standardni model /k-e, standard model A 368,84 90,49 371,87 91,61
k-e, Kato-Launder A 368,98 90,72 372,12 91,83
k-e, RNG, Kato-Launder A 369,03 90,67 372,26 91,82
k-e, RNG, Kato-Launder B 369,10 90,67 372,53 91,85 k-w, standardni model /k-w, standard model C 369,99 90,88 372,75 91,89 k-w, Kato-Launder C 370,08 91,03 373,03 92,04 k-w, Kato-Launder, SST C 370,15 91,13 373,11 92,14
Preglednica 3. Energijske izgube v toku in koeficient rekuperacije tlaka v sesalni cevi, dobljene z razliènimi turbulentnimi modeli na redki in zgoèeni mrei
Table 3. Flow-energy losses and coefficient of pressure recovery in the draft tube obtained by several turbu-lence models on coarse and refined grids
A - standardne log. stenske funkcije / standard log.-law wall functions
C - kombinacija stenskih funkcij za nizka in visoka Re tevila / combined low and high Re wall functions 170 000 vozlov
170 000 nodes 452 000 vozlov 452 000 nodes Turbulentni model
Turbulence model Model za tok ob steni Near-wall model DE/g
m C% p DE/gm C% p
k-e, standardni model, k-e, standard model A 0,257 51,02 0,251 51,9 k-e, Kato-Launder A 0,2528 52,02 0,2421 53,43
k-e, RNG, Kato-Launder A 0,2667 51,21 0,2689 50,39 k-w, standardni model /k-w, standard model C 0,2616 49,31 0,2535 51,58 k-w, Kato-Launder C 0,2665 49,53 0,2588 50,33 k-w, Kato-Launder, SST C 0,272 47,52 0,273 48,37
Preglednica 4. Izkoristek turbine v optimalni toèki obratovanja za razliène turbulentne modele in za dve gostoti mre
Table 4. Turbine efficiency at the best-efficiency point for different turbulence models and for coarse and refined grids
A - standardne log. stenske funkcije / standard log.-law wall functions
C - kombinacija stenskih funkcij za nizka in visoka Re tevila / combined low and high Re wall functions
Turbulentni model
Turbulence model Model za tok ob steni Near-wall model hredke mree /hopt coarse grids
h/hopt goste mree refined grids k-e, standardni model / k-e, standard model A 0,9126 0,9292
k-e, Kato-Launder A 0,9159 0,9332
k-e, RNG, Kato-Launder A 0,9146 0,9313
k-w C 0,9158 0,9311
k-w, Kato-Launder C 0,9182 0,9349
than those obtained by the k-w model. With the Kato-Launder change of production term the flow-energy losses in the case of the k-e model are reduced, while in case of the k-w model they increase. With the k-w SST model the largest flow-energy losses and the smallest Cp are obtained.
From the flow-energy losses in all the turbine parts and from the torque on the shaft, the turbine effi-ciency was calculated. The effieffi-ciency values, divided by the measured efficiency at the best-efficiency point, are presented in Table 4. For all the turbulence models used the efficiency obtained on the refined grid is from 1.5% to 1.7% higher than those obtained on the coarse grids. The highiest efficiency is obtained with the k-w
SST model and the smallest with the standard k-e model.
4 ANALYSIS OF THE FLOW IN A TURBINE WITHOUT MEASURED DATA
All the results presented so far were obtained from calculations where the discharge coresponding to a certain guide-vane opening and speed was obtained from measurements. When a new turbine is being developed, the discharge for a certain operating point is not known: we know only the avaible head and speed. Therefore, we have to solve the problem in a different way. At the turbine inlet the direction of the flow and the total pressure is prescribed, while at the outlet, the static pressure is prescribed. During the calculation, the discharge, which corresponds to the difference in pressure, is calculated. When the spiral casing is calculated separately, the procedure is a bit more compli-cated, because we do not know, how much of the head coresponds to the spiral casing and how much to the other turbine parts. The procedure becomes iterative. For the first approximation it can be assumed that the flow-energy losses in the spiral casing are 1% of the avaible head. Then we prescribe the inlet and outlet boundary conditions for the other parts of the turbine in such a way that the difference in the total pressure corresponds to 99% of the avaible head. A discharge is obtained as a result of a numerical analysis. For this discharge flow analysis of the spiral casing is made and we sum up the difference in the pressure in the spiral casing and in the other turbine parts. If the sum is not equal to the avaible head, we change the boundary conditions for the analysis of the flow from the stay-vane inlet to the draft-tube outlet. In a few steps, we obtain the discharge for which the calculated head is equal to the avaible head. The flow-energy losses in the spiral casing increase quadratically with the discharge, therefore, the flow in the spiral casing is calculated only once. For a new discharge only the flow-energy losses are recalculated.
In this way the flow at five operating points for the nominal head was calculated. The calculation was made with the Kato-Launder k-e model. The cal-culated and measured discharge are compared in Fig. 6. For small guide-vane openings the calculated dis-charge is smaller than the measured one, while for large guide-vane openings it is larger. The largest descrepancy is 1.9%. The calculated efficiency is Launderjevim popravkom dobimo pri modelu k-e
manje izgube, pri modelu k-w pa se izgube celo poveèajo. Z modelom k-w SST dobimo najveèje izgube in najniji Cp.
Z upotevanjem izgub v posameznih delih turbine in navora na os turbine smo za oba dvoenaèbna modela na redki in gosti mrei izraèunali izkoristek turbine. Izkoristek, deljen z izmerjenim izkoristkom v optimalni toèki obratovanja, je prikazan v preglednici 4. Pri vseh modelih se z zgoèanjem mree izkoristek poveèa za od 1,5 do 1,7%. Najveèji izkoristek dobimo z modelom k-w SST, najmanjega pa s standardnim modelom k-e.
4 IZRAÈUN TOKA V TURBINI BREZ PODATKOV IZ MERITEV
Pri vseh do sedaj prikazanih rezultatih je bil pretok pri doloèenem odprtju vodilnika in vrtljajih podatek, dobljen iz meritev. Kadar pa razvijamo novo turbino, tega podatka nimamo. Poznamo le razpololjivi padec in vrtljaje, pri katerih bo turbina obratovala. Zato se v tem primeru naloge lotimo drugaèe. Na vstopu v turbino podamo le smer toka in totalni tlak, na izstopu pa statièni tlak. Med izraèunom se oblikuje pretok, ki ustreza dani tlaèni razliki. Èe raèunamo spiralo posebej, je postopek nekoliko bolj zapleten, saj ne vemo, koliken del tlaène razlike pomenijo izgube v spirali in koliken del odpade na preostalo turbino. Postopek postane iterativen. Za prvi pribliek vzamemo, da izgube v spirali pomenijo 1% celotnega razpololjivega padca. Vstopne in izstopne robne pogoje za preostali del turbine predpiemo tako, da ustrezajo preostalim 99% razpololjivega padca. Kot rezultat numeriène analize toka dobimo neki pretok. Za ta pretok izraèunamo tok v spirali in izgube v njej. Setejemo tlaèno razliko v spirali in v preostali turbini. Èe se ta setevek razlikuje od razpololjivega padca, ustrezno popravimo vstopne in izstopne pogoje za izraèun toka od predvodilnika do izstopa iz sesalne cevi. Po nekaj korakih se v okviru predpisane natanènosti pribliamo razpololjivemu padcu. Ker se izgube v spirali veèajo s kvadratom pretoka, tok v spirali raèunamo le enkrat, nato pa le e preraèunamo izgube, ki ustrezajo novemu pretoku.
1,9%. Izraèunani izkoristek je za okoli 2,5% manji od izmerjenega, oblika krivulje se dobro ujema z meritvami (sl. 7).
5 SKUPNI IZRAÈUN TOKA V CELOTNI TURBINI
Pri dosedanjih izraèunih smo raèunali spiralo posebej, pri kaskadah predvodilnika, vodilnika in gonilnika pa smo se omejili na en periodièni del. S tem smo predpostavili, da je tok med poljubnima lopaticama enak. To zlasti v primeru predvodilnika ne dri. V obravnavanem primeru celo lopatice predvodilnika niso enake, ampak se njihova dolina manja od vstopnega dela proti ostrogi spirale. Da bi ugotovili, koliken je vpliv loèenega izraèuna toka v spirali in vpliv omejitve kaskad na en periodièni del, smo izraèunali tok v celotni turbini od vstopa v spiralo do izstopa iz sesalne cevi z vsemi predvodilnimi, vodilnimi in gonilnimi lopaticami. Zaradi premajhnih raèunalnikih zmogljivosti je uporabljena raèunska mrea zelo redka, ima okoli 510 000 vozlov. Za primerjavo rezultatov v optimalni toèki obratovanja smo tudi loèen in sklopljen izraèun ponovili na enako redkih mreah. Primerjali smo izraèunane izkoristke, deljene z izkoristkom v optimalni toèki obratovanja. Z loèenim izraèunom vsakega dela turbine smo dobili vrednost 0,851, z loèenim izraèunom spirale in skupnim izraèunom toka v preostalih delih 0,929, s skupnim izraèunom toka v celotni turbini pa 0,9398. Izkoristek pri skupnem izraèunu toka v celotni turbini je najveèji predvsem zaradi manjih izgub v kaskadah predvodilnika in vodilnika. Te izgube so se zmanjale zaradi skupnega izraèuna spirale in kaskade. Pri izraèunih, pri katerih smo se omejili na en periodièni del, smo vzeli najdaljo predvodilno lopatico in smo tudi zato dobili prevelike izgube.
about 2.5% less than the measured one (Fig. 7). The shape of the calculated efficiency curve is in good agreement with the measurements.
5 COUPLED CALCULATION OF THE FLOW IN A COMPLETE TURBINE
In all the calculations so far, the flow in the spiral casing was calculated separately and the stay vane, the guide vane and the runner cascades were reduced to one periodical part. It was assumed that the flow between any two blades or vanes is equal. However, this is not true, especially for stay vanes. The stay vanes are not equal, their length decreases from the inlet part to the nose of the spiral casing. To see effect of a separate calculation of the spiral casing and the effect of the reduction of the cas-cades domains to one periodical part, the flow in the whole turbine, from the spiral casing inlet to the draft-tube outlet with all the stay and guide vanes and all the runner blades was calculated simultaneously. Due to insufficient com-puter capacity the grid is very coarse, it consists of about 510 000 nodes. To compare the results at the best-effi-ciency point, the separate and coupled calculation was also repeated on equally coarse grids. The calculated val-ues of efficiency, divided by the measured efficiency at the BEP were compared. By a separate analysis of each tur-bine part a value of 0.851 was obtained, by a separate analysis of the spiral casing and coupled analysis of the other turbine parts we got a value of 0.929, and by a simul-taneous calculation of the whole turbine a value of 0.9398 was obtained. The efficiency was the highest for a simulta-neous calculation of the flow in the whole turbine, mainly due to the smaller flow-energy losses in the stay- and guide-vane cascades. The reason for the smaller losses in the tandem cascade is the coupled calculation of the spiral casing. In the calculation where the domains were reduced to one periodical part the flow between the longest two stay vanes was calculated and that was also the reason for the too high flow-energy losses.
- izraèun - calculation
D - meritve measurement
Sl. 6. Izmerjeni in izraèunani pretok pri dani geometrijski obliki, padcu in vrtljajih
Fig. 6. Measured and calculated discharge for a given geometry, head and speed
- izraèun - calculation
D - meritve measurement
Sl. 7. Izraèunani in izmerjeni izkoristek pri dani geometrijski obliki, padcu in vrtljajih Fig. 7. Calculated and measured efficiency for a
given geometry, head and speed
0,25 0,3 0,35 0,4
0,7 0,8 0,9 1 1,1 1,2 1,3 1,4
A0
Q (
m
3 /s)
0,9 0,92 0,94 0,96 0,98 1
0,16 0,18 0,2 0,22 0,24 0,26 jj
hh
//hh
6 SKLEP
Iz prikazanih rezultatov lahko povzamemo, da z loèenim izraèunom toka v spirali in s skupnim izraèunom toka v dvojni kaskadi, gonilniku in sesalni cevi dovolj natanèno napovemo lego optimalne toèke obratovanja, tudi oblika diagrama izkoristka se dobro ujema z izmerjeno. Izraèunani izkoristek je za okoli 3% manji od izmerjenega, vendar se z zgostitvijo mre izmerjenim rezultatom zelo pribliamo. Pri loèenem izraèunu smo z zgostitvijo mre in vkljuèitvijo Kato-Launderjevega popravka v model k-e dobili za 2% viji izkoristek. Enako izboljanje lahko prièakujemo tudi pri sklopljenem izraèunu na gosti mrei, ki pa zaradi premajhnih raèunalnikih zmogljivosti ni bil izveden. Sklepamo, da se s sklopljenim izraèunom na zgoèeni mrei s Kato Launderjevim modelom k-e izmerjenemu izkoristku lahko pribliamo na 1%.
S skupnim izraèunom toka v celotni turbini na zelo redki mrei smo dobili za 1% veèji izkoristek kakor pri loèenem izraèunu spirale in skupnem izraèunu toka v preostalih delih. S tem smo pokazali, da je izraèunani izkoristek v prejnjih izraèunih premajhen tudi zaradi loèenega izraèuna toka v spirali in omejitve kaskad na en periodièni del. Zato lahko bolj kakor na 1% natanène vrednosti izkoristka prièakujemo le z izraèunom celotne turbine na zelo gosti mrei.
V postopku razvoja novih gonilnikov je pomembno, da tudi v primeru, ko ne poznamo pretoka pri danem odprtju vodilnika, tega lahko dokaj natanèno izraèunamo. Tudi v tem primeru se lega optimalne toèke obratovanja in oblika diagrama izkoristka dobro ujemata z izmerjenimi rezultati.
6 CONCLUSION
From the results presented it can be con-cluded that by a separate analysis of the flow in a spiral casing and a coupled analysis in a tandem cas-cade, runner and draft tube, the position of the best-efficiency point and the shape of the best-efficiency dia-gram are accurately predicted. The calculated effi-ciency is about 3% lower than the measured one, but with grid refinement the calculated results approach the measured ones. With grid refinement and by in-cluding the Kato-Launder production term in the k-e
model the efficiency of the separate analysis in-creased by 2%. The same improvement can be ex-pected for a coupled calculation on a refined grid. This calculation was not performed due to computer capacity. We would expect that the efficiency ob-tained with a coupled calculation on a refined grid would be within 1% of the measured value.
The efficiency obtained from a simultaneous calculation of the flow in a complete turbine with a very coarse grid is 1% higher than that obtained by a separate analysis of the spiral casing and a coupled analysis of the flow in the other turbine parts. This result shows that the calculated efficiency is also too low because of a sepa-rate analysis of the spiral casing and the reduction of cascades domains to one periodical part. Therefore, less than a 1% difference between the calculated and mea-sured efficiency can be expected only with a coupled calculation of the whole turbine on a very fine grid.
In the design process it is very important that when the discharge coresponding to a certain guide-vane opening is not known, it can be calculated accu-rately enough. Also, in this case the position of the best-efficiency point and the shape of efficieny dia-gram is in good agreement with measured results.
7 LITERATURA 7 REFERENCES
[1] Jot, D., A. Lipej, K. Oberdank, M. Jamnik, B. Velenek (1996) Numerical flow analysis of a Kaplan turbine;
Hydraulic Machinery and Cavitation, ed. E Cabrera, V. Espert, F. Martinez, Dortrecht: Kluver.
[2] Troha, J., M. Bajd, A. Oberdank, A. Lipej, D. Jot (1997) Refurbishment and uprating hydro powerplants with model test; Hydropower into the next Century, Portoro, 1997; The International Journal on Hydro-power & Dams; Sutton.
[3] Sick, M., M.V. Casey, P.F. Galpin (1996) Validation of a stage calculation in a Francis turbine; Hydraulic Machinery and Cavitation, ed. Cabrera, E et all, Vol. I.
[4] Jot, D., L. kerget (2000) Separate and coupled CFD simulation of a flow in a Francis turbine; Hydro, Techology and Enviroment for New Century, Hydraulic Machinery and Systems, 20th IAHR Symposyum,
Charlotte, USA.
[5] Jot, D., L. kerget (1999) Numerièna analiza toka v francisovi turbini, Kuhljevi dnevi 99, Slovensko drutvo za mehaniko.
[6] Yakhot, V., S.A. Orszag, S. Tangham, T.B. Gatski, C.G. Speciale (1992) Development of turbulance models for shear flows by a double expansion technique; Phys. Fluids, Volume 7, 1510-1520.
[8] Menter, F. R. (1993) Multiscale models for turbulent flows; 24th Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics.
[9] Menter, F. R. (1996) A comparison of some recent eddy - viscosity turbulence models; Journal of Fluids Engineering, ASME, Vol. 118, 514-519.
[10] Kato, M., B.E. Launder (1993) The modelling of turbulent flow around stationary and vibrating square cylinder; 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, 10-4-1 - 10-4-6.
[11] Grotjans, H., F.R. Menter (1998) Wall functions for general application CFD codes; ed. Papailiou, ECOMAS 98 Proceedings of the Fourth European Computational Fluid Dynamics Conference, 1112-1117. [12] CFX-TASCflow Computational Fluid Dynamics Software, Primer Documentation, Version 2.10.
Naslova avtorjev: mag. Dragica Jot Turboinstitut Rovnikova 7
1210 Ljubljana entvid
prof. dr. Leopold kerget Fakulteta za strojnitvo Univerza v Mariboru Smetanova ulica 17 2000 Maribor
Authors Addresses: Mag. Dragica Jot Turboinstitute Rovnikova 7
1210 Ljubljana entvid
Prof. Dr. Leopold kerget Faculty of Mechanical Eng. University of Maribor Smetanova ulica 17 2000 Maribor, Slovenia
Prejeto: 22.1.2001 Sprejeto: 27.6.2001
